Abstract
This paper presents a new kind of uniform spline curve, named trigonometric polynomial B-splines, over spaceΩ = span{sint, cost,t k−3,t k−4, …,t, 1} of whichk is an arbitrary integer larger than or equal to 3. We show that trigonometric polynomial B-spline curves have many similar properties to traditional B-splines. Based on the explicit representation of the curve we have also presented the subdivision formulae for this new kind of curve. Since the new spline can include both polynomial curves and trigonometric curves as special cases without rational form, it can be used as an efficient new model for geometric design in the fields of CAD/CAM.
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References
Farin, G., Curves and Surfaces for Computer Aided Geometric Design, New York: Academic Press, 1988, 1–273.
Piegl, L., Tiller, W., The NURBS Book, 2nd ed., Berlin: Springer, 1997.
Pottmann, H., Wagner, M. G., Helix splines as example of affine Tchebycheffian splines, Advan. in Comput. Math., 1994, 2: 123–142.
Mainar, E., Peńa, J. M., Sánchez-Reyes, J., Shape preserving alternatives to the rational Bezier model, Computer Aided Geometric Design, 2001, 18: 37–60.
Pottmann, H., The geometry of Tchebycheffian spines, Computer Aided Geometric Design, 1993, 10: 181–210.
Zhang, J. W., C-curves: an extension of cubic curves, Computer Aided Geometric Design, 1996, 13: 199–217.
Zhang, J. W., Two different forms of C-B-Splines, Computer Aided Geometric Design, 1997, 14: 31–41.
Mazure, M. L., Chebyshev-Bernstein bases, Computer Aided Geometric Design, 1999, 16: 649–669.
Wagner, M. G., Pottmann, H., Symmetric Tchebycheffian B-spline schemes, in Curves and Surfaces in Geometric Design (eds. Laurent, P. J., Le Mehaute, A., Schumaker, L. L.), Natick, MA: AK Peters, 1994, 483–490.
Schumaker, L. L., Spline functions: Basic Theory, New York: Wiley, 1981, 363–499.
Piegl, L., Tiller, W., Curve and surface construction using rational B-splines, Computer Aided Design, 1987, 19: 487–498.
Lane, J. M., Riesenfeld, R. F., A theoretical development for the computer generation and display of piecewise polynomial surfaces, IEEE Transaction on Pattern Analysis and Machine Intelligence, 1980, PAMI-2(1): 35–46.
Gordan, W. J., Riesenfeld, R. F., B-spline curves and surfaces, Computer Aided Geometric Design, 1974, 95–126.
Morin, G., Warren, J., Weimer, H., A subdivision scheme for surfaces of revolution, Computer Aided Geometric Design, 2001, 18: 483–502.
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Lü, Y., Wang, G. & Yang, X. Uniform trigonometric polynomial B-spline curves. Sci China Ser F 45, 335–343 (2002). https://doi.org/10.1007/BF02714091
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DOI: https://doi.org/10.1007/BF02714091